Entropy, stability, and Yang–Mills flow

Kelleher, Casey Lynn; Streets, Jeff

doi:10.1142/s0219199715500327

Cited by 11 publications

(11 citation statements)

References 17 publications

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“…Many results in this paper have also been obtained by Kelleher and Streets [17]. Many results in this paper have also been obtained by Kelleher and Streets [17].…”

Section: Zhengxiang Chen and Yongbing Zhangsupporting

confidence: 68%

Stabilities of homothetically shrinking Yang-Mills solitons

Chen

Zhang

2015

Trans. Amer. Math. Soc.

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In this paper we introduce entropy-stability and F-stability for homothetically shrinking Yang-Mills solitons, employing entropy and the second variation of the F -functional respectively. For a homothetically shrinking soliton which does not descend, we prove that entropy-stability implies F-stability. These stabilities have connections with the study of Type-I singularities of the Yang-Mills flow. Two byproducts are also included: We show that the Yang-Mills flow in dimension four cannot develop a Type-I singularity, and we obtain a gap theorem for homothetically shrinking solitons.

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“…Many results in this paper have also been obtained by Kelleher and Streets [17]. Many results in this paper have also been obtained by Kelleher and Streets [17].…”

Section: Zhengxiang Chen and Yongbing Zhangsupporting

confidence: 68%

Stabilities of homothetically shrinking Yang-Mills solitons

Chen

Zhang

2015

Trans. Amer. Math. Soc.

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“…These objects correspond to solutions of the YM heat flow on the trivial bundle over R d , which is our main motivation to study the problem in this geometrical setting. Moreover, Weinkove [34] as well as by Chen and Zhang [9]. However, to the best of our knowledge no rigorous proof on the stability of the Weinkove solution given in Eq.…”

Section: 4mentioning

confidence: 94%

“…The results of Weinkove have raised interest in the stability of YM-solitons in recent years and notions of variational stability have been introduced by Kelleher and Streets [34] as well as by Chen and Zhang [9]. However, to the best of our knowledge no rigorous proof on the stability of the Weinkove solution given in Eq.…”

mentioning

confidence: 99%

Stable blowup for the supercritical Yang–Mills heat flow

Donninger

Schörkhuber

2019

J. Differential Geom.

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In this paper, we consider the heat flow for Yang-Mills connections on R 5 × SO(5). In the SO(5)−equivariant setting, the Yang-Mills heat equation reduces to a single semilinear reactiondiffusion equation for which an explicit self-similar blowup solution was found by Weinkove [57]. We prove the nonlinear asymptotic stability of this solution under small perturbations. In particular, we show that there exists an open set of initial conditions in a suitable topology such that the corresponding solutions blow up in finite time and converge to a non-trivial self-similar blowup profile on an unbounded domain. Convergence is obtained in suitable Sobolev norms and in L ∞ . 7 9

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“…As a result, it is not strong enough to control the small scale behaviour of a G 2 -structure ϕ. In this section, motivated by analogous functionals for the mean curvature flow [CM12], the high dimensional Yang-Mills flow [KS16] and the Harmonic map heat flow [BKS17], we introduce an entropy functional, and use it and the almost monotonicity of Section 5.1 to establish an ǫ-regularity result, as well as to show that small entropy controls torsion.…”

Section: Entropy and ε-Regularitymentioning

confidence: 99%

A Gradient Flow of Isometric $$\mathrm {G}_2$$-Structures

2019

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We study a flow of G2-structures that all induce the same Riemannian metric. This isometric flow is the negative gradient flow of an energy functional. We prove Shi-type estimates for the torsion tensor T along the flow. We show that at a finite-time singularity the torsion must blow up, so the flow will exist as long as the torsion remains bounded. We prove a Cheeger-Gromov type compactness theorem for the flow. We describe an Uhlenbeck-type trick which together with a modification of the underlying connection yields a nice reaction-diffusion equation for the torsion along the flow. We define a scale-invariant quantity Θ for any solution of the flow and prove that it is almost monotonic along the flow. Inspired by the work of Colding-Minicozzi [CM12] on the mean curvature flow, we define an entropy functional and after proving an ε-regularity theorem, we show that low entropy initial data lead to solutions of the flow that exist for all time and converge smoothly to a G2-structure with divergence-free torsion. We also study finite-time singularities and show that at the singular time the flow converges to a smooth G2-structure outside a closed set of finite 5-dimensional Hausdorff measure. Finally, we prove that if the singularity is of Type-I then a sequence of blow-ups of a solution admits a subsequence that converges to a shrinking soliton for the flow.

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Entropy, stability, and Yang–Mills flow

Cited by 11 publications

References 17 publications

Stabilities of homothetically shrinking Yang-Mills solitons

Stabilities of homothetically shrinking Yang-Mills solitons

Stable blowup for the supercritical Yang–Mills heat flow

A Gradient Flow of Isometric $$\mathrm {G}_2$$-Structures

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